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Math Help - Chebyshev's inequality

  1. #1
    Member roshanhero's Avatar
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    Chebyshev's inequality

    If X is a non-negative random variable for which E(X) exists,show that for every t>0.
    P[X\geq t] \leq\frac {E(X)}{t}
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by roshanhero View Post
    If X is a non-negative random variable for which E(X) exists,show that for every t>0.
    P[X\geq t] \leq\frac {E(X)}{t}
    \frac{E(X)}{t}=\int_0^{\infty} \frac{x}{t}p(x)\; dx = \int_0^t\frac{x}{t}p(x)\; dx + \int_t^{\infty}\frac{x}{t}p(x)\; dx

    ......... \ge \int_t^{\infty}\frac{x}{t}p(x)\; dx

    and as \frac{x}{t}\ge 1 for x\ge t hence:

    \frac{E(X)}{t}\ge \int_t^{\infty}\frac{x}{t}p(x)\; dx\ge \int_t^{\infty}p(x)\; dx

    CB
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  3. #3
    Member roshanhero's Avatar
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    How can we prove chebyshev inequality from this relation?
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    MHF Contributor matheagle's Avatar
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    And you didn't notice that under Mr Fantasy's link...
    http://people.csail.mit.edu/ronitt/COURSE/S07/lec25.pdf
    under your last post...
    http://www.mathhelpforum.com/math-he...nequality.html
    it's Theorem ONE, on page ONE.
    Last edited by matheagle; July 16th 2009 at 06:48 AM.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by roshanhero View Post
    How can we prove chebyshev inequality from this relation?
    Put X=(Y-\mu)^2

    CB
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  6. #6
    Member roshanhero's Avatar
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    Thanks,but,what should i use in t.
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by roshanhero View Post
    Thanks,but,what should i use in t.
    try it, see what you get then refer th Chebyshev's inequality and see what you have to do with t to get the required form.

    CB
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